This paper is devoted to a generalisation of the quantum adiabatic theorem to a nonlinear setting. We consider a Hamiltonian operator which depends on the time variable and on a finite number of parameters, defined on a separable Hilbert space with a fixed basis. The right hand side of the nonlinear evolution equation we study is given by the action of the Hamiltonian on the unknown vector, with its parameters replaced by the moduli of the first coordinates of the vector. We prove existence of solutions to this equation and consider their asymptotics in the adiabatic regime, i..e. when the Hamiltonian is slowly varying in time. Under natural spectral hypotheses, we prove the existence of instantaneous nonlinear eigenvectors for the Hamiltonian, and show the existence of solutions which remain close to these time-dependent nonlinear eigenvectors, up to a rapidly oscillating phase, in the adiabatic regime. We first investigate the case of bounded operators and then exhibit a set of spectral assumptions under which the result extends to unbounded Hamiltonians.

中文翻译：

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非线性量子绝热近似

本文致力于将量子绝热定理推广到非线性环境。我们考虑一个依赖于时间变量和有限数量参数的哈密顿算子，它定义在具有固定基的可分离希尔伯特空间上。我们研究的非线性演化方程的右侧是由哈密顿量对未知向量的作用给出的，其参数由向量的第一个坐标的模代替。我们证明了这个方程的解的存在性，并在绝热状态下考虑它们的渐近性，即 当哈密顿量随时间缓慢变化时。在自然谱假设下，我们证明了哈密顿量的瞬时非线性特征向量的存在，并证明了与这些时间相关的非线性特征向量保持接近的解的存在，在绝热状态下达到快速振荡阶段。我们首先研究有界算子的情况，然后展示一组谱假设，在这些假设下，结果扩展到无界哈密顿量。